- Romil Jain romilj@cse.yorku

1. Introduction to Google PageRank Algorithm - Romil Jain romilj@cse.yorku.ca

2. World Wide Web • WWW is HUGE. Approximate estimations [1]: • ~50 million active web sites • ~25 billion web pages • ~1 billion users • There are a large number of search engines too [2]: • At least 3,105 search engines

3. Crawler Module Ranking Module Query Module Indexing Module Page Repository Indexes Results Anatomy of a Search Engine User Query WWW

4. Ranking Module • Key is to find those pages that the user desires • Takes a set of relevant web pages and ranks them • Rank is generally a function of: • Content Score & • Popularity Score (The focus of this talk) • E.g. “What are some good Indian restaurants in Toronto?”

5. r(Pj)  r(Pi) : PageRank of page Pi Bi : set of pages pointing to Pi | Pj | : # out-links from Pj r(Pi) = |Pj| Pj  Bi Ranking Web Pages by Popularity • PageRank algorithm, given by Sergey Brin and Larry Page in 1998 [1] • Exploits the linked structure of the web for computing popularity

6. k : kth iteration rk(Pj)  rk+1(Pi) = |Pj| Pj  Bi Ranking by Popularity (cont’d) r(Pj)  • But r(Pj) are unknown ! • So use and iterative procedure: r(Pi) = |Pj| Pj  Bi • r0(Pj) = 1/n, where n is # web pages

7. Example 1 2 3 6 5 4

8. P1 P2 P3 P4 P5 P6 P1 0 1/2 1/2 0 0 0 P2 0 0 0 0 0 0 Hyperlink Matrix H = P3 1/3 1/3 0 0 1/3 0 P4 0 0 0 0 1/2 1/2 P5 0 0 0 1/2 0 1/2 P6 0 0 0 1 0 0 rk(Pj)  rk+1(Pi) = |Pj| Pj  Bi r0(Pj) = 1/n Matrix Notation 1 2 3 6 5 4 (k+1)T = (k)TH,  (k)T : PageRank vector after kth iteration  (0)T : 1/n eT

9. (k+1)T = (k)T H Nice (?) Properties of H • Sparse n n matrix • Less storage space (25 billion web pages!) • Each iteration requires  (nnz(H)) computations. H has about 10n nonzero. So  (n) computations. • Note that a dense matrix would require  (n2) computation • The dangling nodescreate 0 rows in H. All other rows have sum = 1. Thus H is substochastic matrix

10. (k+1)T = (k)T H 1 2 with (0)T = (1 0),(k)T will flip-flop between (1 0) and (0 1) ! Issues with Iterative Process • Will it converge or continue indefinitely? • What properties of Hwill ensure convergence? • Does convergence depend on (0)T ? • How long will it take to converge i.e. what k is the fixed point? • Does a converged T give useful page ranks? All these questions can be answered using theory of Markov Chains & Stochastic Matrices…

11. Stochastic Matrix A stochastic matrix S is: • n n matrix with each row-sum = 1 • for each sij ,0  sij1 Markov Chain for a Random Surfer Transition Probability Matrix

12. Power of Stochastic Matrix If we start from C, what is the probability that we will reach B in 2 steps? P(CB2) = P(CA)P(AB) + P(CB)P(BB) + P(CC)P(CB)

13. It can be proven for a stochastic matrix S that: lim Sn = S* , if 0  sij  1 n Power Convergence In 3, 4, 5, 6, 7 steps?

14. State Vector Transition If xTis a stochastic probability distribution vector of a given state, then: x (k+1)T= x (k)TS Similar to (k+1)T = (k)T H, except that His not stochastic!

15. x (n+1)T = x (n)T S State Vector Convergence If we start with x(0)T, then lim x(n)T = x (0)T lim Sn = x (0)TS* = x* n n

16. P1 P2 P3 P4 P5 P6 P1 0 1/2 1/2 0 0 0 P2 0 0 0 0 0 0 Hyperlink Matrix H = P3 1/3 1/3 0 0 1/3 0 P4 0 0 0 0 1/2 1/2 P5 0 0 0 1/2 0 1/2 P6 0 0 0 1 0 0 The problem is due to these dangling rows H is not stochastic! (k+1)T = (k)T H

17. P1 P2 P3 P4 P5 P6 P1 0 1/2 1/2 0 0 0 P2 1/6 1/6 1/6 1/6 1/6 1/6 P3 1/3 1/3 0 0 1/3 0 P4 0 0 0 0 1/2 1/2 P5 0 0 0 1/2 0 1/2 P6 0 0 0 1 0 0 Dangling rows eliminated… S = Adjustment 1 to H A random surfer can randomly “jump” to any page after he encounters a dangling node S = H + a(1/n eT) a is called the dangling node vector. ai = 1 if page i is dangling otherwise 0.

18. G = S + (1 - ) E , 0    1 E = 1/n eeT is called the teleportation matrix  is the % of time a user surfs or teleports G is called the Google Matrix (k+1)T = (k)T S Adjustment 2 to H 0  sij  1 not true for S! A random surfer can randomly “teleport” to any page irrespective of the current page.

19. Finally we have G! G = S + (1 - ) E , 0    1 (k+1)T = (k)T G • Gis stochastic • 0  gij  1 true for G Therefore the above equation converges for any (0)T But now G is no longer sparse . In fact it is completely dense!

20. (k+1)T = (k)T G Fortunately… G = S + (1 - ) E = S + (1 - ) 1/n eeT = (H + 1/n aeT) + (1 - ) 1/n eeT = H + (a + (1 - ) e) 1/n eT Therefore: (k+1)T = (k)T G =  (k)T H + ( (k)T a + (1 - ) (k)T e ) 1/n eT =  (k)T H + ( (k)T a + (1 - )) 1/n eT (?) Now vector multiplications are done on extremely sparse H

21. (k+1)T = (k)T G Importance of  G = S + (1 - ) E , 0    1 (k+1)T = (k)T G What  must be chosen? It can be shown that rate of convergence is the rate at which k  0   0, T converges immediately, but completely unrealistic!   1, Tmay never converge, again unrealistic ! We want  to be as close as possible to 1

22. (k+1)T = (k)T G  = 0.85 Saves the Day G = S + (1 - ) E, 0    1 Brin & Page initially chose  = 0.85, and this is still the value used by Google Takes about 50 iterations (3 days) to converge sufficiently Accuracy is 50= .8550 .000296, which is sufficient for Google’s needs

23. (k+1)T = (k)T G Importance of Teleportation Matrix E G = S + (1 - ) E Initially we had E = 1/n eeT This means that a random surfer can teleport to any web page with equal probability 1/n Instead of 1/n eeT use evT , where vTis the personalization or teleportation vector. vT is used to counter-act link farms (like SearchKing.com)

24. (k+1)T = (k)T G Issue: Sensitivity of PageRank It can be shown that: 1 d (k)T  d  1 -  as   1, 1/(1- )  So, PageRank is quite sensitive to small changes in the web. Google computes PageRank from scratch every month! Can we compute i+1 from i without computing i+1 from scratch?

25. (k+1)T = (k)T G Issue: PageRank is Query Independent! • PageRank is pre-computed. • It means that to be better linked is more important than to contain the search terms • This is significant because a badly linked page, might be popular within the community of pages with the same topic A rosy idea: Is it feasible to compute PageRank after the relevant documents have been retrieved?

26. (k+1)T = (k)T G Issue: PageRank is Dead! Not for now, but is susceptible to a lot of damage: • PageRank is based upon an ideal democratic structure of the web • But hackers, spammers and SEO’s know too much about Google to skew the rankings • Typical examples are Link Farms and Google Bombs. • Bloggers created a bomb where if you typed “miserable failure” then Google would take you to www.whitehouse.gov! How can we detect and fight Rank Skewing?

27. References • The size of the World Wide Web, May 2007. http://www.pandia.com/sew/383-web-size.html. • Search Engines Worldwide, Jan 2003. http://home.inter.net/takakuwa/search/search.html . • Langville and Meyer. Google’s PageRank and Beyond. Princeton University Press, 2006. • Brin and Page. The Anatomy of a Large-scale Hypertextual Web Search Engine. Computer Networks and ISDN Systems, 1998.